Edexcel IGCSE Further Pure Maths answers

Answers Chapter 1

4

a

625

b

9

c

7

d

2

5

a

1.30

b

0.602

c

3.85

d

20.105

1.1 Exercise 1A

6

a

1.04

b

1.55

c

20.523 d

2

3

a e i m q a d g a

x7 k5 2a3 27x8 4a6 x5 x3 1 _ 3x 2 65

b f j n r b e h b

6x5 y10 2p27 24x11 6a12 x22 x5 5x 69

e

6 _3 1

f

6 ___ 64

j

125

i

c g k o

2p2 5x8 6a29 63a12

3x22 p2 3a2b22 32y6

d h l p

c f i c

x4 12x0 = 12 6 x21 3 d

1 ___ 125

g

1

h

66

9 _ 4

k

5 _

l

64 __

6

__

1 __ 16

9 7√2

__

12 9√5

2

__

13 23√5

14 2

√5 16 ___ 5__ √ 5 19 ___ 5___ √ 13 22 ____ 13

√ 11 17 ____ 11 1 20 __ 2 1 23 __ 3

__

__

11 23√7

15 19√3 __

___

√2 18 ___ 2 1 21 __ 4

2

3

a c a c e a e

log4 256 5 4 log10 1 000 000 5 6 241 = 16 _ 92 = 3 105 = 100 000 3 b 2 1 6 f _2

3 4

d a c e

e

log10 120 b log6 36 = 2 1 d log8 2 = _3

a b c

3 loga x 1 4 loga y 1 loga z 5 loga x 2 2 loga y 2 1 2 loga x

d

loga x 1 _2 loga y 2 loga z

e

1 1 _ 1 _ log

log2 8 = 3 log12 144 = 2 log10 10 = 1

a d a a a d a

c g

2

a

2

2.460 0.458 1.27 1 _ 2 , 512 6.23 1.66 1 _ 2 , 512

3.465 0.774 2.09 1 _ 1 __ 16 , 4 2.10

c

b

1 _ 1 __ ,

c 2.52

4.248

c 0.721 c 0.431

16 4

y x

x

y=6

y=4

x

(b)

1

d h

b e b e b

y = ( 41)

log3 (_9 ) 5 22 log11 11 5 1 52 = 25 521 = 0.2 7 21

log5 80

1

1

b d b d

c

1.6 Exercise 1F

1.3 Exercise 1C 1

log2 9

1

8 6√5

10 12√7

b

68 log6 (__ 81 )

__

7 √3

__

log2 21

1.5 Exercise 1E

√2 3 5__ 6 √3

__

a

__

2 6√___ 2 5 3√10 __

3

49

1 2√__ 7 4 4√2 __

1 2

1.2 Exercise 1B __

1.4 Exercise 1D

Edexcel IGCSE Further Pure Mathematics

1

3.00

(a) 1 (c)

1 10

(4  )

1 NB __

x

2x

=4

x

2x

so (c) is y = 4

1

2

x

y = ( 31 )

7

y y=3

(c)

x

So x = log580

10 _______ 10

y = log3x

i.e. x = 2.72270 …

(b)

= 2.72 3sf

x

1


(=  loglog 805 )

(a) 1

7x = 123

b ⇒

x = log7123

(=  loglog 123 7 ) 10 ________

x

10

NB y = log3x is a reflection of y = 3 in the line y = x. x 1 y = __ is y = 32x 3

i.e. x = 2.47297 …

(  ) y

3

8

y = log4x

y = log3x

1 3 5 7 9 11 13 15 17 19 21 23

y = lx

1

x

1 (b)

y = 1x = 1 y = log3x = 1 ⇒ x = 31 = 3 So coordinates of intersection are (3, 1)

Exercise 1G a

y8

2

a

3x6

3

a

4

a

1

5

a

6

a

b

1 _ ,9

2 4 6 8 10 12 14 16 18 20 22 24

2x(x 1 3) (x 1 6)(x 1 2) (x 2 8)(x 2 2) (x 2 6)(x 1 4) (x 1 5)(x 2 4) (3x 2 2)(x 1 4) 2(3x 1 2)(x 2 2) 2(x2 1 3)(x2 1 4) (x 1 7)(x 2 7) (3x 1 5y)(3x 2 5y) 2(x 1 5)(x 2 5) 3(5x 2 1)(x 1 3)

2 4 6 8 10

(x 2 3)2 2 9 1 (x 1 _2 )2 2 _1 4 2(x 1 4)2 2 32 2(x 2 1)2 2 2 5 25 2(x 2 _4 )2 2 __

9

2.1 Exercise 2A

y

a c

c

Chapter 2

(a)

4

= 2.47 3sf b 12

9

3

x

(b)

a

1 9 _3 , 9 10 2_1 , 22

y = log6x

1

(a)

2

5x = 80

a

6x7

b 62 3375 4 __ b _____ 9__ 4913 __ √7 ___ b 4√5 7__ √3 15 ___ __ b ___ 3 √5 2 logd p + logd q

c

32x

d

c

6x2

d

12b9 1 _31 __ x 2

x(x 1 4) (x 1 8)(x 1 3) (x 1 8)(x 2 5) (x 1 2)(x 1 3) (x 2 5)(x 1 2) (2x 1 1)(x 1 2) (5x 2 1)(x 2 3) (2x 2 3)(x 1 5) (x 1 2)(x 2 2) (2x 1 5)(2x 2 5) 4(3x 1 1)(3x 2 1) 2(3x 2 2)(x 2 1)

2.2 Exercise 2B 1 3 5 7 9

(x 1 2)2 2 4 (x 2 8)2 2 64 (x 2 7)2 2 49 3(x 2 4)2 2 48 5(x 1 2)2 2 20

11 3(x 1 _2 )2 2 __ 4 3

b

loga p = 4, logd q = 1

27

8 1

12 3(x 2 _6 )2 2 __ 12 1

1 3 5 7 9 11

x 5 0 or x 5 4 x 5 0 or x 5 2 x 5 21 or x 5 22 x 5 25 or x 5 22 x 5 3 or x 5 5 x 5 6 or x 5 21

2 4 6 8 10 12

x 5 0 or x 5 25 x 5 0 or x 5 6 x 5 21 or x 5 24 x 5 3 or x 5 22 x 5 4 or x 5 5 x 5 6 or x 5 22

1 13 x 5 2 _2 or x 5 23 3 2 15 x 5 2 _ or x 5 _

3 1 14 x 5 2 _3 or x 5 _2 3 5 16 x 5 _ or x 5 _

17 x 5 _3 or x 5 22

18 x 5 3 or x 5 0

19 x 5 13 or x 5 1

20 x 5 2 or x 5 22

√5 21 x 5 6 ___

22

3

2

1

__

3

___

√ 11 23 x 5 1 6 ____ 3

2

7

__

27 x 5 23 6 2√2 ___

26 x 5 0 or x 5 2 __ 62 11

___

28 x = 26 ± √33 __

29 x 5 5 6 √30

30 x = 22 ± √6

3 √29 31 x 5 __ 6 ____ 2 2

3 __ 32 x = 1 ± __√2 2

1 √129 33 x 5 __ 6 _____ 8 8

34 No real roots

3 √39 35 x 5 2 __ 6 ____

4 √ 26 36 x = 2 __ ± ____

___

____

___

2

___

5

5

1

x2 2 2x + 1 = 0 ⇒ (x 2 1)2 = 0 2

__

___

+3 6 √17 10 _________ , 20.56 or 3.56 2 __

11 23 6 √3 , 21.27 or 24.73 ___

5 6 √ 33 12 ________ , 5.37 or 20.37 2 ___ √ 31 13 5 6 ____, 23.52 or 0.19 3 __

1 6 √2 14 _______ , 1.21 or 20.21 2

14___ 22 6 √19 16 __________ , 0.47 or 21.27 5 ___

21 6 √78 18 __________ , 0.71 or 20.89 11

so equal roots x51

so two real roots __ __ √ 2 ± 8 x 5 ______ = 1 ± √ 2 or 2.41, 20.414 3sf 2 3 b2 2 4ac = (23)2 2 4(22) 5 17 so two real roots

2.5 Exercise 2E x2 + 5x + 2 = 0 a + b = 25 ab = 2

1

___

4

2

___

√ 53 15 9 6 ____, 20.12 or 21.16

1

b2 2 4ac = (22)2 2 4(21) 5 8

3 ± √17 x 5 ________

2x2 = x 1 4 = 0 b2 2 4ac = (21)2 2 4 3 (2) 3 (24) = 33 so two real roots ___ √ 1 ± 33 x = ________ 4 x = 1.69 or 21.19 3sf

17 2 or 2 _4

2.4 Exercise 2D b2 2 4ac = (22)2 2 4 3 1 5 0

⇒

2x2 2 x 2 4

8

√5 9 23 6 ___, 20.38 or 22.62 2

___

x 5 3 6 √13

7

1

⇒

3x2 + x 2 7

2

24 x 5 1 or x 5 2 _6

25 x 5 2 _2 or x 5 _3

2

3x2 = 7 2 x = 0 b2 2 4ac = 12 2 4 3 3 3 (27) = 1 + 84 = 85 so two real___ roots 21 ± √85 _________ x = 6 x = 1.37 or 21.70 3sf

7


2.3 Exercise 2C

= 3.56 or 20.562 3sf

b2 2 4ac = (23)2 2 4 3 4 5 9 2 16 = 27 so no real roots

a

2a + 1 + 2b + 1 = 2(a + b) + 2 = 210 + 2 = 28 (2a + 1)(2b + 1) = 4ab + 2(a + b) + 1 = 8 2 10 + 1 = 21  new equation is x2 1 8x 2 1 5 0

b

ab + a2b2 = ab(1 + ab) = 2(1 + 2) = 6 (ab)(a2b2) = (ab)3 = 23 = 8  new equation is x2 2 6x 1 8 5 0

5 b2 2 4ac = (1)2 2 4 3 2 3 (22) 5 17 so two real roots ___

21 ± √17 x 5 _________ = 0.781, 21.28 3sf 4 6 b2 2 4ac = (21)2 2 4 3 3 3 3 5 235 so no real roots

3

___

x2 + 6x + 1

2

=0 a + b = 26 ab = 1

a


b

4

(a + 3) + (b + 3) = (a + b) + 6 = 0 (a + 3)(b + 3) = ab + 3(a + b) + 9 = 1 2 18 + 9 = 28 2  new equation is x 2 8 5 0 b a2 + b2 (a + b)2 2 2ab a __ __ + = _______ = ______________ b a ab ab 36 2 2 _______ = = 34 1 b a __ __ 3 =1 b a  new equation is x2 2 34x 1 1 5 0 x2 2 x + 3

3

=0 a+b=1 ab = 3

a

b

4

a

b

a

25 ± √17 _________ , 20.44 or 24.56

b

2 ± √ 7 , 4.65 or 20.65

c

23 ± √29 _________ , 0.24 or 20.84

d

2

__

___

10 ___ 5 ± √73 ________ , 2.25 or 20.59 6

a

6

64

7

a

ab 5 t, a2 1 b2 5 2t(2t 2 1)

b

√ 577 t 5 1 1 _____ 2

c

x2 2 2√577 x 1 1 5 0

____

____

2x2 2 7x 1 3 = 0

8

x2 2 _2 x + _2 = 0 7

7

ab = _2 3

a2 + b2 = (a + b)2 2 2ab = 12 2 6 = 25 a2 3 b2= (ab)2 = 32 = 9  new equation is x2 1 5x 1 9 5 0 x2 1 x 2 1 = 0 a + b = 21 ab = 21 1 b + a 21 1 __ __ + = _____ = ____ = 1 a b ab 21 1 __ 1 ___ 1 ____ 1 __ 3 = = = 21 a b ab 21  new equation is x2 2 x 2 1 5 0 b a+b a _____ ___ + = _____ = 1 ab a + b a + b

3

a

_ __ a2 + b2 = (a + b)2 2 2ab = __ 4 2232= 4

b

a 2 b = √(a 2 b)2 = √ a2 + b2 2 2ab

49

________

4

37

3

_____________

___

________

_ __ _ = √__ 4 2 2. 2 = √ 4 = 2 37

c

3

25

5

a3 2 b3 = (a 2 b)(a2 + b2 + ab) _ _ __ ___ = _2 ( __ 4 + 2) = 2 3 4 = 8 5 37

3

5

43

215

x2 2 2tx + t = 0

9 a

a + b = 2t ab = t a2 + b2 = (a + b)2 2 2ab = 4t2 2 2t = 2t(2t 2 1)

b

a + b = 2t a 2 b = 24 ⇒ 2a = 2t + 24 a = t + 12, b = t 2 12  ab = t ⇒ (t + 12)(t 2 12) = t

a c a c e

x(3x 1 4) x(x 1 y 1 y2) (x 1 1)(x + 2) (x 2 7)(x 1 5) (5x 1 2)(x 2 3)

b d b d f

2y(2y 1 5) 2xy(4y 1 5x) 3x(x 1 2) (2x 2 3)(x + 1) (1 2 x)(6 + x)

a

y 5 21 or 22

b

x 5 _3 or 25

c

1 x 5 2 _ or 3

d

√7 5 6 ___

5

3

a + b = _2

(a + 2) + (b + 2) = (a + b) + 4 = 1 + 4 = 5 (a + 2)(b + 2) = ab + 2(a + b) + 4 =312+4=9  new equation is x2 2 5x + 9 5 0

Mixed Exercise 2F 2

b

p 5 3, q 5 2, r 5 27

b ab a 21 _____ 3 _____ = _______ = ____ = 21 a + b a + b (a + b)2 1 2  new equation is also x 2 x 2 1 5 0

1

__

√7 22 6 ___ 3

5

2

__

2

i.e. t2 2 144 = t or 0 = t2 2 t 2 144 ________

____

1 + √ 577 1 ± √ 1 + 576  t = ____________  t = _________ (t > 0) 2 2 c

2

2

b a +b 2t(2t 2 1) a __ __ + = _______ = _________ = 2(2t 2 1)

b a ab t b a __ 3 __ = 1 b a  equation is x2 2 2(2t 2 1) x + 1 = 0 ____

or x2 2 2√ 577 x + 1 = 0

Chapter 3

3.3 Exercise 3C

1

2

a c e a c e

x2 1 5x 1 3 x2 2 3x 1 7 x2 2 3x – 2 6x2 1 3x 1 2 2x2 2 2x 2 7 25x2 1 3x 1 5

b d

x2 1 x2 9 x2 2 3x 1 2

b d

3x2 1 2x2 2 23x2 1 5x 2 7 + 6x2

2x3

3

a

b

2x3 + 5x2 2 5x + 1

2x2

=

x2 + 3x 2 1

3x3 + 2x2 2 3x 2 2

=

(3x 1 2)(x2 2 1)

=

x2 2 1

=

(3x 2 1)(2x2 1 x 2 2)

=

2x2 1 x 2 2

3

2

(3x + 2)

6x3 + x2 2 7x 1 2 3

2

6x + x 2 7x 1 2  ________________ 3x 2 1

d

4x3 + 4x2 1 5x 1 12

2x2

 ___________________ = 2x 1 3 2x3 + 7x2 1 7x 1 2

=

2x3 + 7x2 1 7x 1 2  _________________ = 2x 2 1

3.2 Exercise 3B 1 2 3 4 5

6

7 8 9

1

+3x2

(x 2 1)(x 1 3)(x 1 4) (x 1 1)(x 1 7)(x 2 5) (x 2 5)(x 2 4)(x 1 2) (x 2 2)(2x 2 1)(x 1 4) a (x 1 1)(x 2 5)(x 2 6) b (x 2 2)(x 1 1)(x 1 2) c (x 2 5)(x 1 3)(x 2 2) a (x 2 1)(x 1 3)(2x 1 1) b (x 2 3)(x 2 5)(2x 2 1) c (x 1 1)(x 1 2)(3x 2 1) d (x 1 2)(2x 2 1)(3x 1 1) e (x 2 2)(2x 2 5)(2x 1 3) 2 216 p 5 3, q 5 7

c

26

d

0

1

a

x 5 5, y 5 6 or x 5 6, y 5 5

b

x 5 0, y 5 1 or x 5 _5 , y 5 _5

c

x 5 21, y 5 23 or x 5 1, y 5 3

d

x 5 4_2 , y 5 4_2 or x 5 6, y 5 3

e

a 5 1, b 5 5 or a 5 3, b 5 21

4

1

3

1

2

f u 5 1_2 , v 5 4 or u 5 2, v 5 3 (211, 215) and (3, 21)

16x2

3

(21_6 , 24_2 ) and (2, 5)

2x2 2 x 1 4

4

a

x = 21_2 , y = 5_4 or x = 3, y = 21

b

x = 3, y = _2 or x = 6_3 , y = 22_6

a

x 5 3 1 √ 13 , y 5 23 1 √13 or x 5 3 2 √13 , ___ y 5 23 2 √__13 __ __ x 5 2 2 3√__ 5 , y 5 3 1 2√5 or x 5 2 1 3√5 , y 5 3 2 2√5

22x2

=

a 27 b 27 18 30 29 8 8__ 27 a = 5, b = 28 p 5 8, q 5 3

3.4 Exercise 3D

22x2

4x3 + 4x2 1 5x 1 12

e

(2x 2 1)(x2 + 3x 2 1)

2x3 + 5x2 2 5x + 1  _________________ 2x 2 1

3x + 2x 2 3x 2 2  _________________ c

=

1 2 3 4 6 7 8 9


3.1 Exercise 3A

(2x 1 3)(2x2 2 x 1 4)

1

16x2

(2x 1 1)(x2 1 3x 1 2) 1x2

5

x2 1 3x 1 2

1

b

1

3

1

1

5

1

___

___

___

3.5 Exercise 3E 1

2

3

a

x,4

b

x>7

c

x . 2_2

d g j a d g j a d

x < 23 x . 212 8 x>3 x , 18 x,4 3 x > _4 1 x . 2_2 No values

e h k b e h

x , 11 x,1 1 x . 1_7 x,1 x.3 x . 27

f i

x , 2_5 x < ??

c f i

x < 23_4 2 x > 4_5 1 x < 2 _2

b e

2,x,4 x54

c

2_2 , x , 3

1

3

1

1

5

3.6 Exercise 3F

2

a c e

3,x,8 x , 22, x . 5 1 2 _2 , x , 7

b d f

24 , x , 3 x < 24, x > 23 1 x , 22, x . 2_2

g

1 1 _ , x , 1_

h

x , _3 , x . 2

j l b d

1 2 x , 22_, x . _

i k a c

2

2

23 , x , 3 x , 0, x . 5 25 , x , 2 1 _ 2,x,1

1

2

3

21_2 , x , 0 x , 21, x . 1 1 23 , x , _4 1

13 a Let a = no. of adults, and a + c < 14

c = no. of children. (no more than 14 passengers) (money raised must cover cost of £72) (more children than adults) (at least 2 adults)

12a + 8c > 72 c .a

3.7 Exercise 3G 1 2 3 4 5

23 < x , 4 y , 2 or y > 5 2y 1 x > 10 or 2y 1 x < 4 22 < 2x 2 y < 2 4x 1 3y < 12, y > 0 and y , 2x 1 4 3x 3x 2 3 6 y . < ___ 2 3, y < 0 and y > 2___ 4 2 3x 2x 7 x > 0, y > 0, y , 2 ______ and y < 2 ______ 219 316 8 y > 0, y < x + 2, y < 2x 2 2 and y < 18 2 2x 9

a >2 c a=2

14 P

12 10

+


1

12

a=c

8 6

Q M

4

N

2

4

+

2

6

8

3a + 2c = 18

10

11

6

10

12

14

d

a + c = 14

NB 12a + 8c > 72 requires line 3a + 2c = 18 b To find smallest sized group you need to consider points close to M and N M(2, 6) is 2 adults and 6 children Points close to N are (3, 5) and (4, 5) So the smallest sized group is 8: 2 adults and 6 children or 3 adults and 5 children. c To find the maximum amount of money that can be made you need to consider points close to P and Q P(2, 12) raises 2 3 12 1 12 3 8 = £120 Q(7, 7) is not in the region ( c . a) but (6, 8) is on d (6, 8) raises 6 3 12 1 8 3 8 = £136 So the maximum amount available for refreshments is £64 from taking 6 adults and 8 children

y

28

28

24

24

20

20

+

b

16

R

16

3b + 2a = 40 12 R

S

+

12

8

+

8

4

10a + 14b = 140

28

T

d

4

16

20

24

28

32

x

To find maximum profit drag the profit line towards the edges R, S, T. It will first cross at T, then R and finally S. T (16, 0) gives a profit of £192 R (0, 18) gives a profit of £270 S is not a point giving whole numbers for x and y but the nearby points are (6, 14) and (7, 13) (6, 14) gives a profit of 6 3 12 + 14 3 15 = £282 (7, 13) gives a profit of 7 3 12 + 13 3 15 = £279 So maxmimum profit is £282 from making 6 ornament A and 14 ornament B.

a = no. of machine A b = no. of machine B 4a + 5b <100 (floor area is 100m2) 2a + 3b <40 (no. of operators is 40) Profit would be P = 100a + 140b (this is parallel to 10a + 14b) The maximum profit wil run along RQ Nearest to Point R (0, 13) will raise £1820 Point (1, 12) will raise £1780 Point Q (20, 0) will raise £2000 So maximum profit is from 20 Machine A Let

12

4a + 5b = 100

Let

15

8

+

24

+

16

+

12

+

8

+

4

+

4 Q 20

x = no. of ornament A y = no. of ornament B 3x + 2y <48 (machine time is 48 h) 1.5x + 2.5y <45 (craftsman’s time is 45 h) Profit P = 12x + 15y (e.g. 12x + 15y = 120 is drawn)


+

14

Mixed Exercise 3H 1 x 5 4, y 5 3_2 3 1 2 (3, 1) and (22_5 , 21_5 ) 1

3 x 5 21_2 , y 2 2_4 and x 5 4, y 5 2 _2 1 4 a x . 10_2 b x , 22, x . 7 1

1

1

5 3,x,4 1 6 a x , 2_2 1 b _2 , x , 5

c _2 , x , 2_2 x , 0, x . 1 A = 2, B = 4, C = 25 p 5 1, q 5 3 (x 2 2)(x + 4)(2x 2 1) 7 1 7_4 a p 5 1, q 5 215 a r 5 3, s 5 0 a (x 2 1)(x 1 5)(2x 1 1) 22 218 1

7 8 9 10 11 12 13 14 15 16 17

1

b b b

(x 1 3)(2x 2 5) 13 1__ 27 1 25, 2 _2 , 1

7

__

3 √5 18 2, 2 __ ± ___ 2 2 1 19 _, 3

i

2

y

2

20 A: y . 2x + 2, 4y + 3x + 12 . 0 B: y , 2x + 2, 4y + 3x + 12 . 0 C: y , 2x + 2, 4y + 3x + 12 , 0 D: y . 2x + 2, 4y + 3x + 12 , 0 21 {y > 3 and 2x 1 y < 6} 22


j

y

1 2

2

0

x

2

1 2

a

0

3

b

y

x

1 2

y 2

0

1

x

1 1

c

0

2

d

y

x

1

y

2

Chapter 4 e

4.1 Exercise 4A 1

a

b

y

1

f

y

2 x

0 2

y

x

0

2

y

2 x

0

1

x

0 1

6 0

3 2 10

2

c

x

3

6

6

y

i

3 3

e

x

10

2

y

10

3

f

h

y 3

3 0

3

d

y

y

g

x

1

x

1

0

y

j

y

3 x

1

1 x 0

y

2

x

0

x

2

24

3 0

g

2

8

h

y 1 0

10

4 x

3

1

x

2 0 1

x

b y = x(x + 4)(x 1 1) y

y

x

2

y

1

a y = x(x + 2)(x 2 1)

0

1

x

4

10

x

y

x

0

1

e

d y = x(x + 1)(3 2 x)

y

1

e y = x2(x 2 1)

y

x

0

3 x

0

f y = x(1 2 x)(1 1 x)

y

y

2

y

a

b

y

27 0

x

1

10

x

1

x

0

3

0

g y = 3x(2x 2 1)(2x 1 1) h y = x(x + 1)(x 2 2)

27

y y

c 10 2

0

1

0

3

1

9

x

0 b

x

0

d

x

2

x

x

1 2

y

x

y

0

y 1 8

y

0

x

1

e

4.3 Exercise 4C 1

c

y

8

0

0

x

4.2 Exercise 4B

0

d

x

y

y

a

2

j y = x2(x 2 9)

i y = x(x 2 3)(x 1 3)

1

y

x

1 2

3

x

3


c y = x(x + 1)2

y

y

y

0

x

4

x

x

0

y

2

x

9

2

b

y x

x

0


ii

2

x

c

y

3

x

1

3 iii x(x 1 2) 5 2 __ x i y 2 y

x

y

4

y

x

0

1

x

1

y y

8

x

d

y

x

0

y

3

y

5

ii 3 iii x2 5 (x 1 1)(x 2 1)2 i

x

0

y

x2 (1

x)

x

1

y 3

y

ii

x

x

0

y

e

8

x

2

2 iii x2(1 2 x) 5 2 __ x i y

4.4 Exercise 4D 1 a

0

y

i

y

y

x2

ii 1

0

y = x(x2 2 1) ii 3 iii x2 5 x(x2 2 1) 10

2)

x

0

4

x(x

x

x

y

y

0

2

2

y

y

y

2

y

3

i

1

x

1

1

x

1 iii x(x 2 4) 5 __ x

4

y

x( x

x 4)

2

x

(x

1)(x

1)2

y

i

j y

x( x

i

y

4)

y 0

g

x

y

3

1 iii x(x 2 4) 5 2 __ x i

2 a

y

4

y

y

y

2)3

(x

x2(x

4)

x

0

2 0

x3

ii 3 iii 2x3 5 2x(x 1 2) y

h

x

0

2

x( x

x y 4)

b 3 a

ii 1 iii x(x 2 4) 5 (x 2 2)3 i

x(4

x)

(0, 0); (4, 0); (21, 25) y

y

1

y

2

y

x

x)3

y

x(2x

y

x

0

i

x(1

x3

b 4 a

x

0

2.5

ii

2)

1

y

ii

x

4

x(x


f

5)

(0, 0); (2, 18); (22, 22) y

2

2 iii 2x3 5 2 __ x i y

y

(x

1)(x

1)

y y

x2

1 0

(x

1)3

x

1

x

0

y

x3

b 5 a

(0, 21); (1, 0); (3, 8) y y

ii 2 iii 2x3 5 x2

x2

x

0

y

b

27

x

(23, 9) 11

y

6 a y

x2

11 a

0


y

b 7 a

x(x

2

2)(x

0

1

y

b 12 a

2

0

x2(x

x3

3x2

y

3)

1

Only 2 intersections

y

(x

y

0

3x(x

1)

b 13 a

1

(x2

1)(x

2)

x

2

(0, 2); (23, 240); (5, 72) y

x

1

y 2

1)3

(x

2)(x

x

0 2

y

Only 1 intersection

x2

y

9 a

1

y 0

x

x(x

1)2

Graphs do not intersect y

1 2

y

01

x(x

2

2

y

1

(0, 28); (1, 29); (24, 224)

x

1

y

b

2

y

10

b 10 a

14x

x

3

y

b

4x

y

2

b 8 a

x

(0, 0); (22, 212); (5, 30)

x

y

6x

4

y

3)

(0, 0); (2, 0); (4, 8) y

y

x

3

b

12

y

2x

2)2

x 4x2

1, since graphs only cross once

4.5 Exercise 4E 1

y

O 4

x

8

2)2

y

2

Transformation f(x + 1) gives y

x

1

1 y = _____ x+1

x

1 O

Finally transformation f(x) + 3 gives y

y

y=3

1 y = 3 + _____ x+1

x

1 12

x

O

Asymptotes x = 21, y = 3

x = 1

y

4

1 1 8 y = _____ 2 1 Start with y = __ x22 x

y

x

Transformation f(x 2 2) gives y

x

O

1 12

2

5

x

1 y = _____ x22

y

Finally f(x) 2 1 gives

5

y

2

O

150

x

330

1 y = _____ 2 1 x22

x

1

Asymptotes x = 2, y = 21

1 9 y = 2 + _____

Vertical asymptote is x = 1 (put denominator = 0) Horizontal asymptote is y 5 2 (let x ⇒ ∞) x21

5

6


3

y

y

x = 0, y = 1

2 1

3

O

1

x

Asymptotes x 5 1 and y52 3 + 2x 2 (1 + x) + 1 1 10 y = ______ = ___________ = 2 + _____ 1+x 1+x 1+x

y

1 1 7 y = 3 + _____ Start with y = __ x+1 x

x

y x 1

3 2 O

x

x = 0, y = 3 Asymptotes x = 21, y = 2

13

3 11 y = 2 + _____ 12x

2

Vertical asymptote x = 1 Horizontal asymptote y = 2 Let x = 0 and y = 2 + 3 = 5 y NB This is a series of 2 3 transformations of y = 2 __ O 1 x

y

1)

y 0

1

x

(0, 0)

x

(1, 0)

y

c

1

0

y y=5 x

O

3

y

ln(2x)

x = 3

3 13 y = _____ 2 4 22x

d

Let x = 0 and y = _2 2 4 = 2 2 _3 3

2

This is a series of transformations 3 of y = 2 __ x

x

( 12 , 0)

Vertical asymptote x = 2 Horizontal asymptote y = 24

y = 3ln (x 2 2)

y

x>2

y

2

x

O

Use transformation f(x 2 2)

4.6 Exercise 4F a

b

y

y

e

x

1

y

Now use transformation 3f(x)

y y

4e

2x

y = 3ln (x 2 2)

(0, 4)

1

y

e

x

x

y

x)

ln(4

d

y 2ex

y

ln(4) x

c

y

3

4

(3, 0)

x

4 (0, 3)

y

y

y

e

y

1

10e 2 x

6

16

y

f x

110

(0, 3 y

100e

x

10 x

3

ln(x

x

ln2)

10

2

6

y

2

3

f

y

x 4

x

e

x

1)

(0,

2

x

3

y

4

1

x

1

Start with y = lnx

y

4

(0, 2)

14

2 ln(x)

x

2

ln(x

x

x

Vertical asymptote x = 23 Horizontal asymptote y = 5

Let x = 0 and y = 5 2 _3 = 4_3 Again this is a series of 2 transformations of y = 2 __ x

y

b

+


2 12 y = 5 2 _____ 3+x

y

a

(e

3

2, 0)

x

2)

2

3

x

⇒ _3 ex = 2 2 2x 1

4.7 Exercise 4G 1

a b

 2 + _3 ex = 4 – 2x so draw y = 4 2 2x 1

x 5 1, y 5 4.21; x 5 5, y 5 3.16

3

y

a b

and intersection at  0.65 or 0.7 to 1sf x 5 1, y 5 21ln1 5 2; x 5 4, y 5 3.39 y

+

5

5 y=4

+

y=x

4

+

3

4

5

6

x

0 4

x

+ +

y = 2.5

+

d

+

+

)

y=x

lnx = 0.5 2 y ⇒ 2 + lnx = 2.5 so draw y = 2.5 Intersection  1.60 1 4 x = ex 2 2 ⇒3 lnx =0 x 2 2 1 2 3 4 ⇒ 2 + lnx = x so draw y =x 1 2 Intersection  3.1 to 1sf x5 30, y 5 2.60; x 5 75, y 5 1.98 1 +

c

x

+

+

y

4 +

9

a b

y

0 4

7

1 3

y=2

10

20

30

40

50

60

70

80

0

+

8

y=6

+

2 2

5

90 x

y=2

+

+

6

+

0

20

30

40

50

60

70

80

90 x

0

+

+

1

10

+

+

+

1

4 3

4

+

1 3

3

+

b

2

+

( 

1

+

a

e – = 0.5 ⇒ 3 + 2e – = 3 + 2 3 0.5 = 4 Draw y = 4 and intersection is at  1.35 x22 x = 22 ln _____ 2 x _ x22 ⇒ e– 2 = _____ 2 x 1 _ _ ⇒ 2e– 2 = x 2 2  3 + 2e– 2 x = 1 + x Draw y 5 1 1x and intersection at x  2.55 1 x 5 0, y 5 2 1 _3 = 2.33; x = 2.5, y = 6.06

2

y = 4 – 2x

+

+

2

1 _ 2x

+

d

1 _ 2x

+

3

+

2

+

1

+

1 5

+

2 y

1

c

y = 2.5

+

2

+

y=1+x


+

+

3

+

+

+

+

4

2

c

1

–1

1

2

3

4

5

6

c

ex = 12 ⇒ 2 + _3 ex = 2 + _3 3 12 = 6

d

Draw line y = 6 and intersection is at  2.45 x = ln (6 2 6x) ⇒ ex = 6 2 6x

1

1

x

2 + 2 cosx = 5 sin 2x ⇒ 2 = 5 sin 2x 2 2 cosx Draw y = 2 and intersections at  25.1 and 74.6 or 25 and 75 to 2sf

15

Mixed Exercise 4H a

y

x2(x

y

y 1.0

2)

y= x 4

+

1

b

0.8 0.6

x

0.4

+

2

0

0.2

y

1

y y

x2

a

c

2x

5

d

A(23, 22) B(2, 3) y = x2 + 2x 2 5 y = x2 2 2x 2 3 = (x 2 3)(x + 1)

7

x

ln(x 2 1) = 0 ⇒ 1 2 ln(x 2 1) = 1 Intersection at x = 2

so draw y = 1

x _

x = 1 + e1 + 4

ln(x 2 1) = 1 + _4 x

x _

x

3

1

⇒ x 2 1 = e1 2 4 x x x ⇒ 1 2 ln(x 2 1) = 1 2 (1 1 _4 ) = _4 so draw y = _4 and intersection at ≈ 2.5

3

y

b

y = x2 2 2x 1 4 = (x 2 1)2 + 3

Chapter 5

4 3 x

1

5.1 Exercise 5A

y

4

a

y = 2(x2 2 4x + 3) = (3 2 x)(x 2 1)

1

x

3

1 2

3

y

b

5

a

y = 2(x2 2 4x 1 5) = 2 [ (x 2 2)2 +1 ] = 21 2 (x 2 2)2 y = _2 ex 1 4 1

2

x

1

3 4

5

y 4.5

x

b

6

a

1

y

y = ln(x 1 1) 1 2 2 (x = 0, y = 2) y = 0 ⇒ ln(x + 1) = 22 0.86 ⇒x+1 = e22  x = e22 2 1 ≈ 20.86 x=5 y = 1 2 ln4 = 20.39 x=6 y = 1 2 ln5 = 20.61

Arithmetic sequences are a, b, c, h, l a 23, 2n + 3 b 32, 3n + 2 c 23, 27 2 3n d 35, 4n 2 5 e 10x, nx f a + 9d, a + (n 2 1)d a £5800 b £(3800 1 200m) a 22 b 40 c 39 d 46 e 18 f n

5.2 Exercise 5B

y=4

(x = 0, y = 4.5)

16

6

0.8

6

x

x

0

5

0.6

x

1

y

3

4

0.4

y B

b c

3

+

x 5 0, 21, 2; points (0, 0), (2, 0), (21, 23)

A

2

+

b a

1 0.2

+

2

x2

+


2x

x

2 3 4 5 6 7

a 78, 4n 2 2 b c 23, 83 2 3n d e 227, 33 2 3n f g 39p, (2n 2 1)p h a 30 b 29 d 31 e 221 d56 a = 36, d = 23, 14th term 24 x = 5; 25, 20, 15 1 7 3 _2 , x 5 8

42, 2n 1 2 39, 2n 2 1 59, 3n 2 1 271x, (9 2 4n)x c 32 f 77

1

3 4 5 6

a 820 b 450 d 2294 e 1440 g 21155 h 21(11x 1 1) a 20 b 25 d 4 or 14 (2 answers) 2550 i £222 500 ii £347 500 1683, 32674 £9.03, 141 days

7 8

d 5 2 _2 , 25.5 a = 6, d = 22

2

c f

21140 1425

c

65

1

10

a

2 3 4 5 6 7 8 9

30

∑ (3r + 1)

b

r =1

2 3 4

16

∑ 4(11 − r )

d

r =1

a 45 c 1010 19 49

b d

5.8 Exercise 5H

∑ 6r r =1

210 112

2

3

9 2 6_3

Doesn’t exist Doesn’t exist 1 _ 1 2 r if |r|,|

a c

Geometric r 5 2 Not geometric

b d

Not geometric Geometric r 5 3

3 2 _3 4 20

e

Geometric r 5 _2

f

Geometric r 5 21

g a c e

Geometric r 5 1 135, 405, 1215 7.5, 3.75, 1.875 p3, p4, p5

h b d f

Geometric r 5 _4 232, 64, 2128 1 ___ 1 ____ 1 __ 64 , 256 , 1024 28x4, 16x5, 232x6

5 6 7 8 9

a

3√3

b

9√3

1

__

1

5.6 Exercise 5F 1

10 __

1 a c e g i 2 _ 2 3

5.5 Exercise 5E 1

a 255 b 63.938 (3 dp) 2 c 2728 d 546_3 e 5460 f 19 680 g 5.994 (3 dp) h 44.938 (3 dp) 9 5 _ _ , 2 4 4 264 2 1 5 1.84 x 1019 a £49 945.41 b £123 876.81 a 2.401 b 48.8234 19 terms 22 terms 26 days, 98.5 miles on the 25th day 25 years

r =1

11

c

∑ (3r − 1)

10, 6250 a 5 1, r 5 2 1 ±_8 26 (from x 5 0), 4 (from x 5 10)

5.7 Exercise 5G 1

5.4 Exercise 5D 1

2 3 4 5

a b c d

486, 39 366, 2 3 3n21 100 25 ___ 25 _____ __ 8 , 128 , n 2 1 2 232, 2512, (22)n21 1.610 51, 2.357 95, (1.1)n21

__

b d f h j

Doesn’t exist Doesn’t exist 1 4_2 90 1 1 _ _ 1 1 2x if |x|, 2


5.3 Exercise 5C

2

40 1 __ 5 13_ 3

3

23 __ 99

4 40 m 1 r , 0 because S∞ , S3, a 5 12, r 5 2 _2 __ 2 10 r = ± __ 3

√

Mixed Exercise 5I 1 a b c

Add 6 to the previous term, i.e. Un11 5 Un 1 6 (or Un 5 6n 2 1) Add 3 to the previous term, i.e. Un11 5 Un 1 3 (or Un 5 3n) Multiply the previous term by 3, i.e. Un11 5 3Un (or Un 5 3n21) 17

d


e f

2 3 4 5 6 7 8 9 10

11

12

a b c a 32 a a a a a b a b d a b c d a c

Subtract 5 from the previous term, i.e. Un11 5 Un 2 5 (or Un 5 15 2 5n) The square numbers (Un 5 n2) Multiply the previous term by 1.2, i.e. Un11 5 1.2Un (or Un 5 (1.2)n21) Arithmetic sequences are: a 5 5, d 5 6 a 5 3, d 5 3 a 5 10, d 5 25 81 b 860 £13 780 b £42 198 a 5 25, d 5 23 b 23810 26 733 b 53 467 5 b 45 d=5 b 59 9 _ c 1.5 d 415 11k 2 3 Not geometric b Geometric r = 1.5 1 Geometric r = _2 c Geometric r = 22 Not geometric e Geometric r = 1 0.8235 (4 dp), 10x (0.7)n21 640, 5 3 2n21 24, 4 3 (21)n21 3 1 n21 ___ _ 128 , 3 3 (2 2 ) 4092 b 19.98 (2 dp) 50 d 3.33 (2 dp)

13 a

9

b

c 14 b d

Doesn’t converge 60.72 3.16

d c

15 b 16 a c 17 a 18 a c

200 76, 60.8 367 1 1 1, _3 , 2 _9 0.8 50

c

19 a

2 _2

b

1

333_3 1

3 _ , 22 4

Chapter 6 6.1 Exercise 6A

18

1 2

1 1 8x 1 28x2 1 56x3 1 2 12x 1 60x2 2 160x3

3

2 3 1 1 5x 1 __ 4 x 1 15x

4

1 2 15x 1 90x2 2 270x3

45

8 _

3 16 __ 3

182.25 8.95 3 1024

b d

d 0.876 380

b d

10 0.189 (3sf) c

14

5 6 7 8

a p55 b 210 c 280 1 2 0.6x 1 0.15x2 2 0.02x3, 0.94148, accurate to 5 dp a 220x3 b 120x3 c 1140x3 b = 22

6.2 Exercise 6B 1

2 3 4 6

a b

1 1 6x 1 12x2 1 8x3, valid for all x 1 1 x 1 x2 1 x3, |x|, 1

c d

3 1 1 _2 x 2 _8 x2 1 __ 16 x , |x| , 1 1 1 2 6x 1 24x2 2 80x3, |x| , _2

e

1 2 x 2 x2 2 _3 x3, |x| , _3

f

___ 3 __ 2 1 2 15x 1 __ 2 x 1 2 x , |x| , 10

g

3 1 2 x 1 _8 x2 2 __ 16 x , |x| , 4

h

√2 2x2 1 ..., |x| , ___

1

1

1

5

1

75

5

125

1

5

__

12

2

1 |x| < _ 2

3x 9 27 3 1 1 ___ 2 _8 x2 1 __ 16 x , 10.148 891 88, accurate to 6 d.p. 2 a = ±8, ±160x3 9x 27x2 27x3 1 2 ___ + _____ + _____, x = 0.01, 955.339(1875) 2 8 16

Mixed Exercise 6C 1

a

p 5 16

b

2

a b

1 2 20x 1 180x2 2 960x3 0.817 04, x = 0.01

3

a

n58

4

a b c d

1 + 24x + 264x2 + 1760x3 1.268 16 1.268 241 795 0.006 45% (3 sf)

5

a

1 2 12x 1 48x2 2 64x3, all x

b

1 1 2x 1 4x2 1 8x3, |x| , _2

b

7 8 9

35 __ 8

1

1 2 2x 1 6x2 2 18x3, |x| , _3 x3 x x2 1 2 __ 2 ___ 2 ____ 4 32 128 27 4 3 135 6 ___ 1 2 _2 x2 1 __ 8 x 2 16 x 1

c

6

c

270

x x2 x3 1145 1 + __ 2 ___ + ___, _____ 2 8 16 512 a b c

n 5 22, a 5 3 2108 1 |x| , _3

21890

Chapter 7

7.3 Exercise 7C

7.1 Exercise 7A c

1

5 1 _ a 1 _b

2

1 1 2 _a 2 _b + c

3

3 4

____

c e

 23.9 d2a a1b2d

b d

a1b1c a1b1c2d

2a 1 2b 1 b 2 _2 a 3 _ 2a 2 b Yes (l 2) Yes (l 21) l = _12 , m = 23

b b d b e b

a1b c b 2 3a 2a 2 b Yes (l 4) c Yes (l 23) f l = 22, m = 1

l = _14 , m = 5 l = 4, m = 8_12

d

l = 22, m = 21

5

1

5

6

2 _6 a 1 (l 2 _6 )b 2ma 1 (m 2 l)b l 5 _12 , m 5 _16 7 2 i2a+b ii _3 a 2 _4 b 7 3 2 (_3 l 2 m)a + (_4 2 _4 l 1 m)b = 0

f

13 __

a

2a 1 b

b

1 1 _ a 1 _b

c

3 _ 3 _ a b

e a

1 _ a

d f

2 _8 a 1 _8 b 5 : 3, k 5 35

5

1

2

a b c d e f

3i 2 j, 4i 1 5j, 22i 1 6j i 1 6j 25i 1 7j ___ ___ √___ 40 = 2√10 √37 ___ √ 74

3

a

14 __

4

2a 1 kb

d e

2b 1 1 2 _4 a + _4 b

4

4

2

2

5

3

4

3

d

v 5 _3 , w 5 _3

a

AC = x + y; BE = _3 y 2 x

2

2 _4 a + _4 b 1:3 ___› ___› 1 2 _3 a + b, 2a + 3b, AG = 3EB ⇒ parallel

5

) (  )

1

___›

i

iii

12

9

c

b

1 a 1 _b 2__ 4

1

( 

5 13 212 1 1 ____ ___ √ 10 23

(  ) (  ) (  ) (  ) (  )

ii

b c

1 ___

  ) (221 229

___›

XM 21 3 ___› 210 XZ 5 6 7 v3 8 1 w 210 0 6

b

3

3 1 _ a 1 _b

d

a

13

8

c

b

3

13

8

(  ) (  )

53 1 27 c ___ 25 24 27 or 223

m 5 3, n 5 1 m 5 22, n 5 5

7 6 __ a + __b 10

b

1 2

6 __ 13

 16 ) (21

  3) (12

Mixed Exercise 7E

b 2 a, _6 (b 2 a), _6 a 1 _6 b

b c d a b d e

f g h

___›

a

No No

a

8

___›

OC 5 22a 1 2b, OD 5 3a 1 2b, OE 5 22a 1 b

1

b2a

5

7

2


25

a a c a d a

2

___›

7.4 Exercise 7D

7.2 Exercise 7B 1 2

6

b

√ 569

a c

6

d

a

2 3 4

1

__›

__›

1

BF = v(_3 y – x) 1

___›

__›

AF = x + BF = x + v(_3 y – x) 1

v = _4 3

(  ) (  ) (  )

___ v 1 w 5 4 , √41 5 5 , √___ 2v – w 5 22 29 ___ 1 , √ 50 v – 2w 5 27

19

6

7

a b


(  ) (  ) (  ) 5 Chloe (7   ); Leo (54  ); Max (23  )

___ p 1 q = 5 , √41 4 9 , √___ 3p 1 q = 2 85 2 7 ____ p 2 3q = 216 , √ 305

Chloe: 74 km, 2.9 km/h Leo: 41 km, 2.1 km/h Max: 13 km, 1.2 km/h

3

12

4

4_3

5

2_4

6

1 _

7

26

8

25

1

2

3

4 5 6 7 8

22

e

2 _ 3 1 _ 2

i a e i

4

7 _ 5

29

b

21

f

5 _ 4 1 _ 2

j b f j

25 2 23

a 4x 2 y 1 3 5 0 c 6x 1 y 2 7 5 0 e 5x 2 3y 1 6 5 0 g 14x 2 7y 2 4 5 0 i 18x 1 3y 1 2 5 0 k 4x2 6y 1 5 5 0 y 1 5x 1 3 2x 1 5y 1 20 5 0 1 y 1 2 _2 x 1 7 2 y 5 _3 x (3, 0)

y 5 2x 1 1 y 5 2x 2 3 1 y 5 _2 x 1 12

b d f

g

y 5 2x

h

y 52 _2 x 1 2b

a Perpendicular c Neither e Perpendicular g Parallel i Perpendicular k Neither 1 y 5 2 _3 x 4x 2 y 1 15 5 0 1 a y 5 22x 1 _2

b d f h j l

Parallel Perpendicular Parallel Perpendicular Parallel Perpendicular

b

y 5 _2 x

c

y 5 2x 2 3

d

y 5 _2 x 2 8

5

a

y 5 3x 1 11

b d

6 7

c y 5 _3 x 1 2 3x 1 2y 2 5 5 0 7x 2 4y 1 2 5 0

y 5 2 _3 x 1 __ 3

c

3

d

1 _

2

y 5 3x 2 6

g

1 _ 2

h

2

3

y 5 2x 1 8

22 2 2 _3 2

l d h l

3 _ 2

4 5

2x 2 3y + 24 = 0 1 2 _5

k c g k b d f h j l

3 _ 2

3

0 22 1 2 _2

3x 2 y 2 2 5 0 4x 2 5y 2 30 5 0 7x 2 3y 5 0 27x 1 9y 2 2 5 0 2x 1 6y 2 3 5 0 6x 2 10y 1 5 5 0

6 7

y = _5 x 1 3 2x 1 3y 2 12 5 0

8 9

8 _

2 3 4

1

a

2

20

6

c

3 _

d

2

g

1 _

h

8

l

1 _

5

e

21

f

1 _

i

2 _

j

24 k _3 q2 2 p2 _______ 5q1p q2p

3

m 1 2

b

1 _

7

n

2

2 1

2

1

2

5

y 5 _3 x 24 4

8.4 Exercise 8D 1

8.2 Exercise 8B

y 5 3x 1 7 y 5 24x 2 11 2 y 5 _3 x 2 5

10 6x 1 15y 2 10 = 0

5 9 (_3 , 0) 10 (0, 5), (24, 0)

1 _

4

a c e

8.1 Exercise 8A a

1

8.3 Exercise 8C

Chapter 8 1

1

2

1 1

1

13

y 5 2 _2 x 1 __ 2 3

17

1 4 7 10

11 a

2 5 8 11

10

__

√5

____

√ 113

5c

3 6 9 12

13 ___ 2√10 ___ a√53 ___ d√ 61

5

____

b

√ 106 __

3b√5 __ 2e√5

a

b

c

d

2

a

c

(0, 6); (4, 10) is ratio 3 : 1 3 3 10 1 1 3 6 3 3 4 1 1 3 0 ______________ _____________ , = (3, 9) 4 4 (1, 5); (22, 8) is ratio 1 : 2 (2 3 5 1 1 3 8) 2 3 1 1 1 3 (22) ______________ ________________ , = (0, 6) 3 3 (3, 27); (22, 8) is ratio 3 : 2 2 3 (27) 1 3 3 8 2 3 3 1 3x 3 (22) ________________ _________________ , = (0, 2) 5 5 (22, 5); (5, 2) is ratio 4 : 3 4 3 5 1 3 3 (22) _____________ 4321335 2 ________________ , = (2, 3_7 ) 7 7 416 218 (4, 2); (6, 8) midpoint ______, ______ 2 2 = (5, 5)

( 

(  ( 

( 

)

( 

)

( 

0 1 12 6 1 2 midpoint _______, ______ 2 2 = (6, 4)

(2, 2); (24, 6)

d

(26, 4); (6, 24)

( 

224 216 midpoint ______, ______ 2 2 = (21, 4)

( 

)

b

(0, 14)

2 a

y = 2 _2 x + 4

b

y 5 _7 x 1 __ 7 , y 5 2x 1 12

y 5 2 _2 x + _2 , (1, 1)

b

(9, 3)

4 a

y

b

222

5 a

y

b

(3, 3)

12

3

1

1

8 a

y = _2 x 2 2

9 a

2x 1 y 5 20

3

b (4, 4) b 6

b

y 5 _4 x 1 _4

c

20

b

y 5 _3 x 1 _3

c

2x 1 y 2 16 = 0

1

1

__

____________________

12 a

________

__

AB = √ (4 2 21)2 + (11 2 1)2 = √ 52 + 102 = 5√ 5 __ __ 1 1 .... Area of Δ ABD = __ AB.CD = __ 3 √ 5 3 5√5 2 2 25 ___ = 2 1 1 _ _ y = 3x 1 3

2 8x7

3 4x3

2 __ 1 4 _3 x2 3

_ 1 5 _4 x2 3

2 _ 1 6 _3 x2 3

7 23x24

8 24x25

9 22x23

10 25x26

11 2 _3 x2 3

4 _

12 2 _2 x2 2

13 22x23

14 1

15 3x2

16 9x8

17 5x4

18 3x2

4

1

1

3 _

9.2 Exercise 9B

6 11x 2 10y + 19 = 0 y 5 2 _2 x 5 3

_______

CD = √ 22 + 12 = √ 5

)

y 5 23x 1 14

5 11 __ = 2 __ 12 x + 6 3 3 5 _2 x 2 _2

ΔABD

1 7x6

1 a

2

d

9.1 Exercise 9A

)

26 1 6 4 2 4 midpoint _______, ______ 2 2 = (0, 0)

1

x

L

Chapter 9

Mixed Exercise 8G

1 _

C A

)

c

10 a

B

)

(0, 6); (12, 2)

7 a

or x 1 2y 2 16 = 0 x = 0 ⇒ y = 8  D is (0, 8)

)

1

)

y

b

3 a

( 

2y 2 14 = 2x + 2

8.6 Exercise 8F 1

(21, 1); (4, 11) in ratio 3 : 2 2 3 (21) + 3 3 4 2 3 1 + 3 3 11 is ________________, ______________ 5 5 so c = (2, 7) 11 2 1 10 1 MAB = _______ = ___ = 2  mi = 2 __ 4 2 21 5 2 1 Equation of l is y 2 7 = 2 _2 (x 2 2)


8.5 Exercise 8E

9

4

a

4x3 2 x22

b

2x23

2

a

0

b

3

a

(2_2 , 26_4 )

11_2

b

(4, 24) and (2, 0)

c

(16, 231)

d

(_2 , 4), (2 _2 , 24)

1

1

c

3 _

2x2 2

1

1

1

1

21

4


5

1 _

a

x 22

b

26x23

d

4 3 _ x 2 2x2

e

22 26x24 +__ 2x

f

1 2 _3 1 _ x 2 _x22

g

23x22

i

5x 2 + _2 x2 2

j

3x2 2 2x + 2

k

12x3 1 18x2

l

24x 2 8 + 2x22

a

1

3

2

3

2

3 _

3

1 _

b

2 _ 9

1 _

1

c

p p = 28 sin __4 [ sin __2 = sin 90° = 1 ] = 28

2x24

c

h 3 + 6x22

24

d

9.4 Exercise 9D 4

1

c

9.3 Exercise 9C 1

a b c d e f

2

3 4

d

dy y = e2x ⇒ ___ = 2e2x dx dy 26x y=e ⇒ ___ = 26e26x dx dy x 2 y = e + 3x ⇒ ___ = ex + 6x dx dy ___ y = sin 2x ⇒ = 2 cos 2x dx dy y = cos 3x ⇒ ___ = 23 sin 3x dx dy y = 3 sin 4x + 4 cos 3x ⇒ ___ = dx 12 cos 4x 212 sin 3x

dy y = sin 5x ⇒ ___ = 5 cos 5x dx dy 1 1 1 1 _ b y = 2 sin 2 x ⇒ ___ = 2 3 _2 cos _2 x = cos _2 x dx dy c y = sin 8x ⇒ ___ = 8 cos 8x dx dy 2 2 2 2 d y = 6 sin _3 x ⇒ ___ = 6 3 _3 cos _3 x = 4 cos _3 x dx dy e y = 2 cos x ⇒ ___ = 22 sin x dx dy 5 5 5 5 _ f y = 6 cos 6 x ⇒ ___ = 6 3 _6 sin _6 x = 25 sin _6 x dx dy g y = cos 4x ⇒ ___ = 24 sin 4x dx dy x x x 1 h y = 4 cos ( _2 ) ⇒ ___ = 24 3 _2 sin _2 = 22 sin _2 dx dy y = 2e2x ⇒ ___ = 22e2x @ (0, 2) dx dy p y = 3 sin x ⇒ ___ = 3 cos x @ ( __3 = x ) dx

2

3

p

22

2

2

y = 4e3x ⇒ y′ = 4e3x 3 6x = 24x e3x 8(1 1 2x)3

b

y = 9e32x ⇒ y′ = 9e3 2 x 3 21 = 29e3 2 x

c

y = e26x ⇒ y′ = e26x 3 26 = 26e26x

d

y = ex

a

c d 4

a

b

c

5

3 p p 1  m = 3 cos ( __3 ) = _2 [ cos __3 = cos 60° = _2 ]

 m = 28 sin ( 2 3 __4 )

2

a

b

a

dy p y = 4 cos 2x ⇒ ___ = 28 sin 2x @ ( x = __4 ) dx

y = (1 + 2x)4 ⇒ y′ = 4(1 + 2x)3 3 2 = 8(1 + 2x)3 y = (1 + x2)3 ⇒ y′ = 3(1 + x2)2 3 2x = 6x(1 1 x2)2 1 1 _ _ 2 1 _______ y = (3 +4x) 2 ⇒ y′ = _2 (3 + 4x)2 2 3 4 = ________ √ 3 1 4x y = (x2 + 2x)3 ⇒ y′ = 3(x2 + 2x)2 3 (2x + 2) = 6(x 1 1)(x2 1 2x)2

2

+ 2x

⇒ y′ = ex

2

+ 2x

3 (2x + 2)

x2 + 2x

Remember x is in radians

5

a b

a b c

6

a b

= 2(x + 1) e y = sin(2x + 1) ⇒ y′ = cos(2x + 1) 3 2 = 2 cos(2x 1 1) y = cos(2x2 + 4) ⇒ y′ = 2sin(2x2 + 4) 3 4x ⇒ 24x sin(2x2 1 4) y = sin3x ⇒ y′ = 3 sin2 3 cos x 3 sin2 x cos x y = cos2 2x ⇒ y′ = 2 cos 2x 3 (2sin 2x) 3 2 = 24 sin 2x cos 2x y = x (1 + 3x)5 ⇒ y′ = (1 + 3x)5 + x.5(1 + 3x)4 3 3 = (1 + 3x)4 [1 + 3x + 15x] = (1 + 3x)4 (1 + 18x) 2 3 y = 2x (1 + 3x ) ⇒ y′ = 2(1 + 3x)3 + 2x 3 3(1 + 3x2)3 3 6x = 2(1 + 3x2)2 [1 + 18x2] 3 y = x (2x + 6)4 ⇒ y′ = 3x2 (2x + 6)4 + x3 3 4(2x + 6)3 3 2 = x2 (2x + 6)3 [6x + 18 + 8x] = 2x2 (2x + 6)3 (7x + 9) y = xe2x ⇒ y′ = e2x + x.2 e2x = e2x(1 1 2x) y = (x2 + 3) e2x ⇒ y′ = 2x 3 e2x + (x2 + 3)(2e2x) = e2x(2x 2 x2 2 3) 2 2 x2 y = (3x 2 5) e ⇒ y′ = 3 3 ex + (3x 2 5)ex 3 2x 2 = ex (6x2 2 10x 1 3) y = x sin x ⇒ y′ = sin x + x cos x y = sin2x cos x ⇒ y′ = 2 sin x cos x 3 cos x + sin2x (2sin x)

y′ = sin x (2 cos2x 2 sin2x)

7

a

b

y = ex cos x ⇒ y′ = ex cos x 2 ex sin x

+ x2 3 3 (3x 2 1)2 3 3 when x = 1

= ex (cos x 2 sin x) (x + 1) 3 5 25x 3 1 5x = __________________ y = _____ ⇒ y′ x+1 (x + 1)2 5x + 5 2 5x = ___________ (x + 1)2 5 = ________2 (x 1 1) (3x 2 2) 3 2 2 2x 3 3 2x y = _______ ⇒ y′ = ____________________ 3x 2 2 (3x 2 2)2

m = 2 3 23 + 1 3 9 3 22 = 16 + 36 = 52 b

m = 6e0 + 2e0 = 8 c

d

24 = _________2 (3x 2 2) 3x2 y = _________ ⇒ y′ (2x 2 1)2 (2x 2 1)2 3 6x 2 3x2 3 2(2x 2 1) 3 2 = __________________________________ (2x 2 1)4 6x (2x 2 1)[2x 2 1 2 2x] = ______________________ (2x 2 1)4 26x = _________3 (2x 2 1) 8

9

e2x 3

1 2 2x 12x3 = _________________ = _______ 4x e e2x

x y = ___ ⇒ y′ e2x

b

(x + 1)ex 2 ex 3 1 ex xex y = _____ ⇒ y′ = ________________ = _______ x+1 (x + 1)2 (x + 1)2

c

e y = ___ ⇒ y′

a

sin x y = _____ ⇒ y′ x

b

cos x 3 ex + ex sin x ex y = _____ ⇒ y′ = _________________ cos x cos2x

x

x2

x2

x2

2

x2

7

ex(sin x 1 cos x) = ______________ cos2 x

c

sin2 x

y _____ ⇒ y′ e2x

1

x 3 2xe 2 e 3 1 e (2x 2 1) = _________________ = ___________

x2 x cos x 2 sin x 3 1 = ________________ x2 x cos x 2 sin x = ____________ x2

e2x 3 2 sin x cos x 2 sin2 x 3

p y = 3 sin2x ⇒ y′ = 6 sin x cos x when x = __ 4 1__ ___ 1 ___ m = 6 3 __ 3 √ = 3 2 √2 p y = x cos x ⇒ y′ = cos x 2 x sin x when x = __ 2 p __ p p p __ __ __ m = cos 2 sin = 0 2 3 1 2 2 2 2 p __ =2 2

9.5 Exercise 9E

2 3 4 5 6

2e2x

a

x2

y = (2x 1 3)e2x ⇒ y′ = 2(2x + 3)e2x + 2 3 e2x when x = 0

6x 2 4 2 6x = ___________ (3x 2 2)2

c

y = x2 (3x 2 1)3 ⇒ y′ = 2x (3x 2 1)3

a y 1 3x 2 6 5 0 b 4y 2 3x 2 4 5 0 c y5x a 7y 1 x 2 48 5 0 b 17y 1 2x 2 212 5 0 y 5 28x 1 10, 8y 2 x 2 145 5 0 e y 5 2ex 2 __ 2 1 _ y 5 3e y = x sin x ⇒ y′ = sin x + x cos x @ (p, 0)  m = sin p + p cos p = 2p  equation of tangent: y 2 0 = 2p (x 2 p) or y = p2 2 xp p y = 2 cos2 x ⇒ y′ = 4 cos x (2sin x) @ (__4 , 1) 1 1__ ___ . __ = 22  m = 24.____ √2 √2 1 normal has gradient = __ 2  equation of normal is: p 1 y 2 1 = __ x 2 __ 2 4 or 8y 2 8 = 4x 2 p

( 

2e2x

= _____________________________ e4x 2 sin xe2x (cos x 2 sin x) = _____________________ e4x 2 sin x(cos x 2 sin x) = __________________ e2x

or


c

10 a

)

8y 2 4x = 8 2 p

9.6 Exercise 9F 1 1 _4 x4 1 x2 1 c 2 22x21 + 3x + c

23

5 _

3 2x 2 2 x3 1 c


4 5 6 7

_ 4 _2 _ x 2 4x 2 1 4x 1 c 3

1

3

x4 1 x23 1 rx 1 c t3 1 t21 1 c 1 2 3 2 _2 _ 1t1c 3 t 1 6t

1 1 _ _ 1 8 _2 x2 1 2x 2 2 2x2 2 1 c p 9 ____5 1 2tx 2 3x21 1 c 5x p 10 _4 t4 1 q2t 1 px3t 1 c

11 a b c

1 4 _ x 1 x3 1 c 2

3 2x 2 __ 1 c x 4 3 _ 2 3 x 1 6x 1 9x 1 c

d

2 3 1 _ x 1 _ x2 23x 1 c

e

_ 4 _2 _ x 1 2x 2 1 c

12 a

3

5

2

3

5

∫2 sin 3x dx

b

∫3e4x dx

c

∫2 cos 3x dx

d

∫2e2x dx

13 a b c d

2 = 2 __ cos 3x 1 c 3 3 = __ e4x 1 c 4 2 = __ sin 3x 1 c 3

= 22e2x 1 c x4 5ex 1 4 cos x 1 ___ 1 c 2 22 cos x 2 2 sin x 1 x2 1 C 2 5ex 1 4 sin x 1 __ 1 c x ex 2 cos x 1 sin x 1 C

9.7 Exercise 9G 1 2 3 4 5

6

24

3t2 1 8t 2 5 b 6t 1 8 v 5 6 m/s, a 5 14 m/s2 8 v 5 3t2 2 4t 1 3 b a 5 6t 2 4 9 a 5 2t 1 10 b a 5 14 m/s2 2 v 5 t + 10t + 5 t3 2 ⇒ s 5 __ 3 + 5t + 5t + d; s = 0 when t = 0 ⇒ d = 0 8 2  when t = 2 s = _3 + 20 + 10 5 32_3 10 a 6 2 2t b 2 m/s c v 5 24 + 6t 2 t2 t3 ⇒ s 5 24t + 3t2 2 __ + d 3 27 s = 100, t = 3 ⇒ 100 = 24 3 3 + 3 3 9 2 ___ + d 3 i.e. 100 5 72 + 27 2 9 + d  d = 10 t3  s 5 10 + 24t + 3t2 2 __ 3

7

10t 48 2 32t a 40 1 10t b 70 m/s a 30 2 1t b 0 m/s a a 5 32 b v 5 32t 1 100 ⇒ s 5 16t2 1 100t 1 d; when t = 0, s = 0 ⇒ d = 0  s 5 16t2 1 100t a a 5 232 b v 5 160 2 32t ⇒ s 5 160t 2 16t2 1 d; t = 0, s = 384 ⇒ d = 384  s 5 384 + 160t 2 16t2 c s 5 0 ⇒ 16(t2 2 10t 2 24) = 0 i.e. 16(t 2 12)(t 1 2) = 0  passes through origin when t = 12

a c a a c

9.8 Exercise 9H 1 2 3

5

228 10 9 3 (2 _4 , 2 _4 )

c

(2 _3 , 1 __ 27 ), (1, 0) 5

d

(3, 218), (2 _3 , __ 27 )

e

(1, 2), (21,22)

f

(3, 27)

1

b b

c c

2 _5 12.25

a a a

217 4 1 1 b (_2 , 9 _4 )

1

1 14

7p ___ 3p ___   8 , 2 12 e ) minimum   8 , 12 e ) maximum, (___ (___ __

√

3p __

__

4

7p __ 4

√

9.9 Exercise 9I 8 2 19_3

b d

9_4 21

e

8__ 12

a b

i 2 ln2 8 i _3

ii ii

64 __ p

3

a

A(1, 3), B(3, 3)

b

4

6_3

1_3

5

a

(2, 12)

b

6

3_8

13_3

1

2

a c

3

5

2p 15

1

2

3

1

9.10 Exercise 9J 1 2 3 4

8 _ p 9

6p 15e2 dy y = 5x4 ⇒ ___ dx 3  δy ≈ 20x δx

= 20x3 x = 20x3 3 ____ 200

x [ 0.5% of x = ____ 200 ]

x4 = ___ 10 δy x4 = ___ × 100 = _________4 × 100 = 2% y 10 3 5x

 % change in y dy y = 3x2 ⇒ ___ dx  δy ≈ 6x δx

5

= 6x 6x2 x = 6x 3 ____ = ____ 100 100 δy 6x2 = ___ × 100 = __________ × 100 y 3x2 3 100 = 2%

 % change in y

6

dv 4 For a sphere: V = __ pr3 = __ = 4pr2 3 dr δr ≈ 0.02 cm δv ≈ 4pr2 dr Use r = 1 ⇒ δv ≈ 4p 3 0.02 = 0.25 cm3

c

9 b A is (21, 0); B is (_3 , 9__ 27 ) 10 3x2 cos 3x + 2x sin 3x x cos x 2 sin x 12 _____________ x2 13 b y = 2x + 1 14 a 2(x3 2 2x)ex + (3x2 2 2)ex 56p 15 ____ 5 5

20 m 3 40 m; 800 m2 2000p cm2 40 cm 800 _____ cm2 4+p 27 216 mm2

4 5

10.1 Exercise 10A 1

2

3 4

g

1 a

3 a b

7__ 32 31

j m 5

p a d

9° 90° 225° 26.4° 99.2° 200° 0.479 0.909

2p __

45 p __ 6 5p __ 12 2p __ 3 4p __ 3 11p ___ 6

0.873 2.79

b e h b e

12° 140° 270° 57.3° 143.2°

c f i c f

75° 210° 540° 65.0° 179.9°

b e b

0.156 20.897

c

1.74

c

p __

f

p __

i

5p __

l

10p ___

o

7p __

c f

1.75 5.59

p __

n

18 p __ 4 4p __ 9 3p __ 4 3p __ 2

b e

1.31 4.01

e h k

8 3

8

9

4

3

x ___ 1 2 2_2 2 2x 2 __ 3

3

x 2 x 21 c f9(x) 5 _____ . 0 for all values of x x 4 (1, 4) x px 5 a y 5 1 2 __ 2 ___ 2 4 2 ______ c m2 (0.280 m2) 41p d2v 10 6 b __ c ____2 , 0  maximum 3 dx 2300p 2 e 22_9 % d _______ 27 250 7 a ____ 2 2x x2 b (5, 125)

( 

8 b

a d g a d g a d a d

Mixed Exercise 9L x 5 4, y 5 20 d2y 15 b ____2 5 ___ . 0  minimum 8 dx 7 5 2 (1, 211) and (_3 , 212__ 27 )

13

Chapter 10

Exercise 9K 1 2 3

__

OP = 3; f ″(x) > 0 so minimum when x = ±2√2 (maximum when x = 0


20x4 i.e. δy = _____ 200

__

)

x = ±2√2 , or x = 0

10.2 Exercise 10B 1 a b c

2 3 4 5 6 7 8 9 10

i 2.7

ii 2.025

iii 7.5p (23.6)

2 i 16_3 1 i 1_

ii 1.8

iii 3.6

ii 0.8

iii 2

10p ___ cm

3

3

2p__ 5√ 2 cm a 10.4 cm 7.5 0.8 p a __ 3 6.8 cm a (R 2 r) cm

b

1_4

b

4p cm (6  + ___ 3 )

c

2.43

1

25


10.3 Exercise 10C 1 a 19.2 cm2 c 1.296p cm2 1 e 5_3 p cm2 2 a 4.47 c 1.98 3 12 cm2 4 b 120 cm2 2 5 40_3 cm 6 a 12 7 8.88 cm2 8 a 1.75 cm2 9 10 11 12

b d f b

x 5 23.2, y 5 2.06 __ 4 a 3.19 cm b 1.73 cm (√ 3 cm) c 9.85 cm d 4.31 cm e 6.84 cm (isosceles) f 9.80 cm 5 a 108(.2)° b 90° c 60° d 52.6° e 137° f 72.2° 6 a 23.7 cm3 b 4.31 cm3 c 20.2 cm3 7 a 155° b 13.7 cm 8 a x = 49.5, area = 1.37 cm2 b x = 55.2, area = 10.6 cm2 c x = 117, area = 6.66 cm2 9 6.50 cm2 10 a 36.1 cm3 b 12.0 cm3

6.75p cm2 38.3 cm2 5 cm2 3.96

c 1.48 cm2 b 25.9 cm2

c 25.9 cm2

4.5 cm2 b 28 cm 78.4 cm b 28 cm

10.4 Exercise 10D __

__

1

a

√2 ___

2__ √ 3 d ___ 2 1 g __ 2 __ √2 j 2 ___ 2 __ √ 3 m ___ 3

b e h k n

√3 2 ___

2 __ √ 3 ___ 2 __ √2 2 ___ 2 21 __

2 √3

c

1 2 __ 2

f

2 _2 1

__

i

2 √3 _____

l

21

o

√3

2

__

10.5 Exercise 10E 1 a b c d e f 2 a d 3 a b c

26

x 5 84, y 5 6.32 x 5 13.5, y 5 16.6 x 5 85, y 5 13.9 x 5 80, y 5 6.22 (Isosceles ) x 5 6.27, y 5 7.16 x 5 4.49, y 5 7.49 (right-angled 48.1 b 45.6 48.7 e 86.5 x 5 74.6, y 5 65.4 x 5 105, y 5 34.6 x 5 59.8, y 5 48.4 x 5 120, y 5 27.3 x 5 56.8, y 5 4.37

c f

14.8 77.4

10.6 Exercise 10F 1 a 11.7 cm b 14.2 cm d 63.4° 2 a 18.6 cm b 28.1 cm 3 a 14.1 cm b 17.3 cm 4 a 28.3 cm b 34.6 cm c 19.5° 5 a 4.47 m b 4.58 m d 12.6° e 26.6° 6 a 407 m b 402 m c 13.3° 7 a 43.3 cm b 68.7 cm 8 a 28.9 cm b 75.7 cm 9 a 16.2 cm b 67.9° d 71.6° 10 a 26.5 cm b 61.8° 11 a 30.3° b 31.6° 12 a 36.9° b 828 cm2 13 a 15 m b 47.7° 14 a 66.4° b 32.9° 15 46.5 m 16 a OW = 4290 m, OS = 2760 m b 36.0° c 197 km/h

c

34.4°

c c c

48.6° 35.4° 35.1°

c

29.2°

c

8.57°

c c c

81.2 cm 22.4° 55.3 cm2

c c

1530 cm2 68.9°

c

€91 300

10.7 Exercise 10G 1

a

2

d g a

2

sin u _____

b 5 c 2cos2 A 2 cos u e tan x0 f tan 3A 4 h sin2 u i 1 LHS 5 sin2 θ + cos2 θ + 2 sin θ cos θ = 1 + 2 sin θ cos θ = RHS

c

d e

f

g 3

4

a d g j a d g

5

j a

b

c

d

e

LHS

1 2 cos2 θ sin2 θ sin θ 5 _________ = _____ = sin θ 3 _____ cos θ cos θ cos θ

LHS

= sin θ tan θ = RHS sin x° cos x° sin2 x° + cos2x° 5 ______ + ______ = ______________ cos x° sin x° sin x° cos x° 1 ___________ = = RHS sin x° cos x°

LHS

5 cos2 A 2 (1 2 cos2 A) = 2 cos2 A 2 1

= 2 (1 2 sin2 A) 2 1 = 1 2 2 sin2 A = RHS LHS 5(4 sin2θ 2 4 sin θ cos θ + cos2 θ) + (sin2 θ + 4 sin θ cos θ + 4 cos2 θ) = 5 (sin2 θ + cos2 θ) = 5 = RHS LHS 5 2 2 (sin2 θ 2 2 sin θ cos θ + cos2 θ) = 2 (sin2 θ + cos2 θ) 2 (sin2 θ 2 2 sin θ cos θ + cos2 θ) = sin2 θ + 2 sin θ cos θ + cos2 θ = (sin θ + cos θ)2 = RHS LHS 5 sin2 x(1 2 sin2y) 2 (1 2 sin2x) sin2y = sin2x 2 sin2y = RHS sin 35° b sin 35° c cos 210° tan 31° e cos u f cos 7u sin 3u h tan 5u i sin A cos 3x __ √3 ___ 1 b 0 c 2 __ __ √ √ 2 2 1 ___ ___ e f 2 _2 2 2 __

√3

__

p p 5 sin __ cos u 1 cos __ sin u 6 6 p 5 sin __ 1 u 5 R.H.S. 6

( 

10.8 Exercise 10H 1

2

3

h

3

i

LHS 5 sin A cos 60° 1 cos A sin 60° 1 sin A cos 60° 2 cos A sin 60° 5 2 sin A cos 60° 1 5 2 sin A (_2 ) 5 sin A 5 R.H.S. cos A cos B 2 sin A sin B cos (A 1 B) LHS 5 _____________________ 5 __________ sin B cos B sin B cos B 5 R.H.S. sin x cos y 1 cos x sin y LHS 5 ____________________ cos x cos y cos x sin y sin x cos y _________ _________ 1 5 cos x cos y cos x cos y 5 tan x 1 tan y 5 R.H.S. cos x cos y 2 sin x sin y LHS 5 ____________________ 1 1 sin x sin y 5 cot x cot y 2 1 1 1 5 cot x cot y 5 R.H.S. __ p p LHS 5 cos u cos __ 2 sin u sin __ 1 √3 sin u 3 __ 3 __ √3 1 5 _2 cos u 2 ___ sin u 1 √3 sin u 2__ √3 1 _ ___ sin u 5 2 cos u 2 2

270° 60°, 300° 140°, 220° 90°, 270° 45.6°, 134.4° 2120, 260, 240, 300 2144, 144 150, 330, 510, 690

b d f h j b d f

a

2p, 0, p, 2p

b

4

a b c d e g i

5

a

1

√2

a c e g i a c e

c

__

√3 ___

)

60°, 240° 15°, 165° 135°, 315° 230°, 310° 135°, 225° 2171, 28.63 2327, 232.9 251, 431 4p 2p 2p 2 ___, 2 ___, ___ 3 3 3

7p 5p p 3p 2 ___, 2 ___, __, ___ d 20.14, 3.00, 6.14 4 4 4 4 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, 360° 60°, 180°, 300° 1 1 1 1 22_2 °, 112_2 °, 202_2 °, 292_2 ° 30°, 150°, 210°, 330° 300° f 225°, 315° 90°, 270° h 50°, 170° 165°, 345° 7p p 2 ___, 2 ___ 12 12

b


b

1.48, 5.85

10.9 Exercise 10I 1 2 3

4

a b a b a b c d e f g a b c

30°, 210° 135°, 315° p, 2p 0.59, 3.73 60°, 120°, 240°, 300° 0°, 180°, 199°, 341°, 360° 60°, 300° 30°, 60°, 120°, 150°, 210°, 240°, 300°, 330° 270° 0°, 18.4°, 180°, 198°, 360° 194°, 270°, 346° 17p ____ 23p 5p ____ 11p ____ ___ , , , 12 12 12 12 2p 4p 0.841, ___, ___, 5.44 3 3 4.01, 5.41

27

Mixed Exercise 10J


1

cos2 u sin2 u sin4 3u 1

a b c

2

a

3

a

4

b a b

5

a

6 7 8

4 + tan x tan y = __________

b

1

2 tan x 2 3 2 sin 2u 5 cos 2u ⇒ 2 sin 2u \ cos 2u 5 1 ⇒ 2 than 2u 5 1 ⇒ than 2u 5 0.5 13.3, 103.3, 193.3, 283.3 225, 345 22.2, 67.8, 202.2, 247.8 23p 2p 5p 5p 11p 11p ____ ____ , b ___, ___, ___, ____ 12 12 3 6 3 6

0°, 131.8°, 228.2° 0, p, 2p a Max 1, u = 100°; Min = 21, u = 280°

9

b

Max 1, u = 330°; Min = 21, u = 150°

a

1 i _2

b

23.8°, 203.8°

27 28 29 30 31 32 33 34 35 36

iii √3 \3

Review exercise 5 A 5 5, B 5 2 _2 , C 5 113 5 _ fmin 5 2 ___ 4 ,x52

1 a b 2

228_4 1

y 4 3 2

O

2

3 a 3 4 814 5 a – 6 a

b b

1 i _4 b 2 a 2 _ c 5

2_2 1

3

c

x

4

125_2 1

P 5 126.8°, Q 5 R 5 26.6° 2 _

1 1 ii _3 a 1 _6 b

5 1 iii _3 a 2 _6 b

b 3 a – b 54.5°, 234.5° 1 4_2 m 21.8°, 38.2°,120° a 3.18, 6.69, 13.04 b – c 2.3 d 0.6 (0.5 acceptable) 6 4 2 11 x 2 18x 1 135x 12 23 , p , 2 13 – 7 8 9 10

28

b c e

1 _3 __ 6√ 3 109.5°

c

y 5 2 _5 (x 2 1)

b

1.11, 2.68

b

2, 64

b

–

1

6

2 23 449_5 p

24 a c 25 a 26 a

__

1 ii _2

14 a (3.14), (5, 24) 15 a – b 12 cm d 54.7° 1 16 _2 x% 5 5 17 a–b (23, 2 _3 ), (2, _2 ) d – 18 a 0.253, 2.89 c 1.91, 2.30 19 a 625 c x 5 2, y 5 3 20 – 21 82.8° 22 a 8i 2 j

cos 2u 5 2 cos2 u 2 1 b sin 2u 5 2 sin u cos u – d 0.767, 1.33, 2.86 e 4 26 b 50 c 17 (2, 4), (5, 16) b x < 2, x > 5

a 2i 2 11j 73.9° a – 15 200 1 a r 5 _2 , r 5 23 46.5°, 133.5°

b

13 2 __ i2_j

b

12 791

b

10

5

5

1 _ 2

a a a c e 37 a b c d e f 38 – 39 a c

40 a b 41 a d 42 a 43 a c

2y 1 x 5 25 b (25, 0) c (10, 0) 1 4 m/s2 b 25_3 m 4y 5 x 1 23 b y 5 24x 1 26 16 (23, 38) d 6__ 17 28 12__ 51 22p 1 q 5 28, 3p 1 q 5 18 22, 24 (x 1 2)(x 2 3)(x 2 4) – 12 2 , 2 __ 5

7__ 12 7

2y 5 3x 2 18 156 x2 x x3 1 1 ___ 2 ____2 1 _____3 2p 8p 16p p 5 6 _2 – b – e – b 1 2 2x 2 4x2 0.087%

b d

3y 5 22x 1 51 216p

1

c

– 408 4000 b d

4.76

2.76132 a 5 1, b 5 25, c 5 8

1

44 45 46 47

–

3 _ 8

65 66 67

3

112

51 52 53 54 55

56 57 58 59

60 61 62 63

O

68

x

4 m/s2 b 90 m 1 1 4 _ _ b 2 a b 2b 2 _3 a c – 2 3 dy a ___ 5 10x cos 3x 2 15x2 sin 3x dx dy 3e3x(x2 1 3) 2 2x e3x b ___ 5 ___________________ dx (x2 1 3)2 0.212 m/s a 1.39 b 28.7° p , 25, p . 2 a 1, 3.75, 5.89, 6.92 b – c 0.79 d 2.1 7 7 9 9 _ _ a A 5 2 2, B 5 2 4 b 2 _4 , x 5 _2 c (1, 4), (7,10) d (2, 0), (5, 0) e – f 24 (22, 1), (21, 3) p2 a i __ 1 6 ii 9 b p 5 64 4 c x2 2 10x 1 9 5 0 9 a 2 __ b 2 11 , 5 c 4 d 16 380 dy a ___ 5 10x e2x 1 2(5x2 2 2)e2x dx dy 2x3 2 x4 1 4x 2 2 b ___ 5 _________________ dx (x 2 x2)2 ln 4 91.1° 2 23__5 x2 x x2 x b 1 1 ___ 2 ___ a 1 1 ___ 2 ____ 12 144 12 72 x x2 c |x| , 4 d 1 1 __ 2 ___ 6 72 e 0.308

48 a 49 a 50

1

cos 2A 5 2 cos2 A 2 1 sin 2A 5 2 sin A cos A – 17.7°, 102.3°, 137.7° __ 3√ 3 ____ e 8 (6, 21), (1,4) a p5 1 5p4qx 1 10p3q2x 1 10p2q3x 1 5pq4x4 1 q5x5 6 12 b p 5 _5 , q 5 __ 5 or p 5 22, q 5 4 a 3 b q 5 20 c a 5 2, b 5 1 d 9 ___ ___ ___ a i √20 ii √40 iii √20 b A 5 90°, B 5 C 5 45° ___ c (5, 5) d √10 660° 67.4° a 2 b log p c r 5 n 2 1, s 5 n d – 12 12 2 a 2x 2 5x 1 2 5 0 b x2 2 ___ x 1 ___ 5 0 p p 8 3 c _3 d _2 24 , p , 3

64 a b c d

69 70 71 72 73

Practice examination papers


e |x| , _6 2y 5 x 2 2 p[_14 e8 1 4e4 1 27_34 ] a – b c 15, 75, 105, 165 d a i y52 ii x 5 21 3 b (0, 3), (2 _2 , 0) c y

Paper 1 1 2 3 4 5

80.4° or 99.6° a – b p 5 210, q 5 33 20 cm2/s x 5 2 y 5 3, x 5 3 y 5 2 4 a p 5 6, q 5 24 b 5i 1 _3 j

6 a 7 a 8 a b 9 a 10 a c 11 a d

__

( _13, 2√_53 )

b 7r __ 3 ___ 1

25 __ p 3

c 3_2 d 11 2 2 a6 1 6a5bx 1 15a4b2x2 1 20a3b3x3 1 15a2b4x4 1 6ab5x5 1 b6x6 4 4 a 5 2 b 5 _3 , a 5 22 b 5 2 _3 5 b 28 c 9, 3 (2, 4) b y 5 4x 2 4 y54 b 8 units2 11.0 cm b 11.9 cm c 40.1° 101.4° e 61.9° 5

b

1

29

Paper 2 1 2e2x sin 3x 1 3e2x cos 3x 2 a 37.0° b 17.2 cm2 2 3 a i y53 ii x 5 2 b i (2_3 , 0) c y

ii (0, 4)

4


3

30

O

4 a

x

0

y

1

2

0.5

x

2 23

1.0

1.5

2.0

2.5

graph drawn 0, 3, 4 1 6 _2

7 a 8 a 9 a d

(ln 3, 36), (0, 4) b 32.02 c 82.2 units2 – b 2.71 c – d 138 – ___ b 2280 c 37 46√37 e 9x2 1 280 1 3 5 0

c i 1.9 – a 5 120x

b b

__

b

–

√3 1 1 __ i _______ √3 2 1

c

2 tan u tan 2u 5 _________ 1 2 tan2 u

d

√2

e

20 __

__

29

21

3.5

4.0

0.649 21.28 24.52 28.61 212.8 215.9 215.9 29.40

b 5 a 6 a

10 a

3.0

ii 1.3 c 11.8 m c 4

__

√3 2 1 __ ii _______ √3 1 1

Edexcel IGCSE Further Pure Maths answers

Recommend Documents